Given two finite posets $P$ and $Q$, we can form the direct product poset $P \times Q$ whose elements are pairs $(p,q) \in P \times Q$ with $(p,q) \leq (p',q')$ if $p \leq p'$ and $q \leq q'$. Let us say a finite, nonempty poset $P$ is indecomposable if $P=P_1 \times P_2$ implies that either $P_1$ or $P_2$ is equal to $P$.

Is it true that a finite, nonempty poset $P$ can be written as a product $P = P_1 \times P_2 \times \cdots \times P_n$ of indecomposable posets in a unique way up to permutation of the factors?

Surely this is a classical question/result. I am apparently having trouble coming up with the right terms to google for, however. I would definitely appreciate any pointer to the literature. If the general result fails, I would also be interested in what mild conditions we can put on $P$ (e.g. gradedness) to guarantee a positive result.